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Exploring Wavelet Techniques��for Filtering and Triggering

John Kelley RU Nijmegen

26 October 2009

Auger Engineering Radio ArrayAERA

Jörg R. Hörandel, Radboud University Nijmegen for

Auger Engineering Radio ArrayReview, Malargüe, April 2009

1.Introduction

2.Physics case

3.Site layout

4.Antennas

5.Electronics

6.Communications

7.Power harvesting and EMC

8.DAQ

9.Risk analysis

10.Management

Previous Work!

•  This work is basically at the same stage… hopefully motivate us to continue

26.10.2009 J. Kelley, Auger NL meeting 2

Ansatz to a wavelet analysis of radio signals

A.M. van den Berg1, J. Coppens

2,3, S. Harmsma

1,3, C. Timmermans

3

November 26, 2007

1KVI, University of Groningen, the Netherlands.

2IMAPP, Radboud University Nijmegen, the Netherlands.

3NIKHEF Nijmegen and Amsterdam, the Netherlands.

1 Introduction

For the identification of radio signals induced by extensive air showers filtering tech-

niques will be needed to increase the signal-to-noise ratio. A denoising technique based

on wavelet analysis is described and applied for a radio event. The radio event used

in this analysis is associated with CDAS event number 3493116 and obtained with the

test setup described in Ref. [1]. The radio event was detected using an Logarithmic

Periodic Dipole Antenna (LPDA) with its associated low-noise amplifier [2], which has

a gain of 21 dB. The antennas are connected to the radio readout system using 160 m

long RG213 cables which have an attenuation at 60 MHz of about 8 dB [3]. To reduce

signals from strong (radio and TV) transmitters, two passive filters were used: a pass

filter (Mini-circuits SBP-60), which has its pass band between 50 and 70 MHz, and

a 70 MHz low-pass filter (Mini-circuits SLP-70). After this filtering stage the signals

were fed into a 31.5 dB flat-band amplifier (Mini-circuits ZKL-2) and digitized using

a 400 MS s−1

12 bits receiver developed by NIKHEF. The present event was recorded

using antenna 1. The LPDA has two planes, one plane directed along the EW di-

rection, the other one along the NS direction. After this initial analysis a ‘random’

sample of events was used to study different digital trigger conditions.

Because of the narrow pass band, a signal in the time domain resembles very much

a sine wave multiplied with a bell-shaped function. The frequency of the sine wave

corresponds to the maximum frequency in the pass filter and the position of the bell-

shaped function in the time domain reflects the (narrow) time position of the induced

pulse. Typically, the length of a pulse is around 10 - 100 ns. As discussed amongst

others by Torrence and Compo [4], a wavelet analysis provides an efficient analysis

method for such time-and-frequency localized pulses, as compared to e.g. a windowed

Fourier transform.

The present document describes a wavelet analysis to the measured pulse, showing

that indeed wavelet analysis can be used to reduce the noise level significantly. The

analysis is based on the software [5] made available by Torrence and Compo [4].

2 Analysis

The wavelet basis used in the present analysis is of the Morlet type. For a wavelet a

scale s can be defined. In the time domain this scale can be regarded as the duration of

the wavelet, i.e. a slow oscillation for a large-scale fluctuation, versus rapid oscillations

for small scales. Mathematically, this function is given as a plane wave multiplied with

1

Why Wavelets?

•  A decomposition of a time series that provides both spectral and temporal information –  Possibly better than DFT for transient analysis

•  Successful use in other fields for weak transient extraction –  Radar reflections (see e.g. Ehara, Sasasi, and Mori 1994) –  GRB light curves (see e.g. thesis by N. Butler, MIT 2003)

•  Transforms can be fast –  Possible implementation in FPGA for trigger

26.10.2009 3 J. Kelley, Auger NL meeting

Discrete Wavelet Transform

•  Basis functions –  finite in time –  scaled and shifted

versions of a “mother wavelet”

•  Decomposition results –  wavelet coefficients –  time on one axis –  period scale (often

factors of 2) on the other

26.10.2009 4

Haar

Daubechies 4

Morlet

J. Kelley, Auger NL meeting

Visualization of Haar DWT


= c00 ×

c10 c11

c20 c21 c22 c23

+

+

+

+ …

c30…c37

Example Radio Pulse (2007)


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-40

-30

-20

-10

0

10

20

30

40

50

s)µTime (2 3 4 5 6 7

) s

P/T

2P

eri

od

scale

(lo

g

2

4

6

8

10

12

-80

-60

-40

-20

0

20

40

60

Event 3388350, pole 2, NS Daubechies4 DWT coefficients

Projection (200 MHz)


Projection (100 MHz)


Projection (50 MHz)


Projection (25 MHz)


Projection (12.5 MHz)


Another Example


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20

Event 3382961, pole 1, NS

TTL noise cosmic

s)µTime (2 3 4 5 6 7

) s

P/T

2P

eri

od

sc

ale

(lo

g2

4

6

8

10

12

-40

-30

-20

-10

0

10

20

30

40

50

coefficients (cij)

Another Example


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20


TTL noise cosmic

power (cij2)

Filtering

•  Hard threshold on wavelet coefficients (or power) can keep pulses but reject white noise

•  Coefficient threshold by Donoho & Johnstone:

•  All coefficients below τ in magnitude set to 0

•  Inverse transform to reconstruct filtered time series 26.10.2009 J. Kelley, Auger NL meeting 17

τ = δ�

lnNsamp (11)

δ =median(|cHF,j |)

0.6745(12)

2

Filtering example


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20

cij threshold = 2 τDonoho

Filtering example


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20

Event 3382961, pole 1, NS cij threshold = 4 τDonoho

s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-30

-20

-10

0

10

20

30

Filtering example


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20


s)µTime (2 3 4 5 6 7

V)

µV

olt

ag

e (

-20

-10

0

10

20

Difficulties

•  Filtering alone doesn’t distinguish between good and bad pulses

•  Transform + filtering + inverse probably too much for L1 trigger in FPGA – Possible exception: short timescale Haar

•  Could be useful for L2 or offline pulse extraction


Trigger?


rolling integral over 25-100 MHz coefficient bins (abs. value)

s)µTime (0 1 2 3 4 5 6 7 8 9 10

V)

µV

olt

ag

e (

-40

-30

-20

-10

0

10

20

30

40

50

s)µTime (2 3 4 5 6 7

) s

P/T

2P

eri

od

scale

(lo

g

2

4

6

8

10

12

-60

-40

-20

0

20

40

60

Rolling Sum

•  Looks promising — but triggering on golden events trivial

•  Will likely still trigger on TTL pulse

•  May be good for power line noise — still looking

•  Don’t know about pulse trains yet


s)µTime (0 1 2 3 4 5 6 7 8 9

| (a

.u.)

ij |

c!

1000

2000

3000

4000

5000

6000

Summary

•  In principle, wavelet analysis is appropriate –  provides temporal and spectral information

•  Threshold filtering can pull pulses out of noise –  could also use for data compression, but impact on spectral

information?

•  Haar wavelet transform feasible in FPGA

•  A wavelet L1 trigger is possible but needs much more work on discrimination power –  ratio in different frequency bands?


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